Rumour Spreading without the Network Alessandro Panconesi - PowerPoint PPT Presentation

Rumour Spreading without the Network Alessandro Panconesi Dipartimento di Informatica Joint work with: Pawel Brach, Alessandro Epasto, Piotr Sankowski

THE ¡STARS ¡

PEOPLE ¡

The ¡INTERNET ¡is ¡an ¡observatory ¡of ¡Crowds ¡

Digital ¡Traces ¡

The ¡Grand ¡Challenge ¡

The Grand Challenge What can we reconstruct the original diffusion process from the huge, and yet scanty, digital traces?

Rumour spreading, a case study

Gossip: a very simple model

Gossiping

Gossiping

Gossiping

Gossiping

Gossiping

Gossiping

Gossiping

Gossiping Variants PUSH Node with information sends to a random neighbour

Gossiping Variants PUSH Node with information sends to a random neighbour PULL Node without information asks a random neighbour

Gossiping Variants PUSH-PULL PUSH Node with information sends to a random neighbour PULL Node without information asks a random neighbour

The problem that we want to solve RUMOUR SPREADING WITHOUT THE NETWORK

Beyond ¡the ¡asymptoBc ¡tradiBon ¡ Can we predict the number of informed nodes at time t on the basis of the degree distribution alone?

Beyond ¡the ¡asymptoBc ¡tradiBon ¡ Can we predict the average number of informed nodes at time t on the basis of the degree distribution alone?

Beyond ¡the ¡asymptoBc ¡tradiBon ¡ Can we predict the average number of informed nodes at time t on the basis of the degree distribution alone for real social networks ?

The Master Plan

The master plan • Develop in a rigorous way a space- efficient simulator for a model • Test it with real networks

THE MODEL

Configuration Model D = ( )

Configuration Model D = ( )

Configuration Model D = ( )

Configuration Model D = ( )

Configuration Model D = ( )

Configuration Model D = ( )

Configuration Model D = ( ) Is this a good model for social networks?

Configuration Model D = ( ) Is this a good model for social networks? No, but this is good!

Problem ¡restatement ¡ Can we predict the average number of informed nodes at time t on the basis of the degree distribution alone for the configuration model ?

Problem ¡restatement ¡ Can we predict the average number of informed nodes at time t on the basis of the degree distribution alone for the configuration model ? YES, OF COURSE!

Naive Simulator • On input D = (d 1 ,d 2 ,…,d n ), pick a random graph G(D) from the configuration model • Pick a random source and simulate rumour spreading • Compute averages • Repeat

THE SPACE-EFFICIENT SIMULATOR

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( )

The Efficient Simulator D = ( ) This is space-efficient because we do not need to keep the stubs, only their number

The Efficient Simulator D = ( ) For undirected networks further optimization is possible. The resulting savings are spectacular

Dealing with aggregates Rank(u) = #unused stubs of node u M[i,j] = #nodes of degree j and rank j DxD matrix

Dealing with aggregates

Theorem • The Efficient Simulator is a correct implementation of the Naïve Simulator-- they compute the same averages

A Picture is Worth a Thousand Words

EXPERIMENTS WITH REAL NETWORKS

Experiments ¡with ¡real ¡networks ¡ ? Input: ¡the ¡degree ¡ Efficient ¡simulator ¡for ¡ distribuBon ¡of ¡a ¡real ¡ the ¡configuraBon ¡model ¡ network ¡

The Good.. Epinions

The Good..

The Bad..

and the Ugly

Different behaviours • Friendship and trust networks: Epinions, Facebook, LiveJournal, RenRen, and Slashdot • Collaboration and Email networks: AstroPh, CondMatt, DBLP and WikiTalk; EuAll and Enron • Non-social newtorks: Web, Amazon

Courtesy of Silvio Lattanzi MEASURING RANDOMNESS

Sudden drops

To summarize • We developed a space-efficient predictor for the configuration model • Surprisingly, this works quite well for real social networks too

Future work • Look for more efficient predictors, eg systems of differential equations • Go beyond averages • Extend to other diffusion processes?

THANKS